The elliptic measure on \(\mathbb {R}^n\)  can be defined as follows. Let \(f\in C_0(\mathbb {R}^n)\)  with \(\operatorname{supp}f\subset \Delta _{R_0}\)  for some \(R_0>0\) . We define an extension of \(f\)  (still denoted by \(f\) ) which is equal to 0 on \(\mathbb {R}^{n+1}_+\setminus T_{R_0}\) . Then for all \(R\ge R_0\) , \(f^\pm \in C(\partial T_R)\) , where \(f^\pm :=\max \left\lbrace \pm f,0\right\rbrace \) . For each \(X\in T_R\) , let \(\omega _R^X\)  be the elliptic measure on \(\partial T_R\) . Then \(u_R^\pm (X):=\int _{\partial T_R}f^\pm d\omega _R^X\)  solves the Dirichlet problem in \(T_R\)  with boundary data \(f^\pm \)  for all \(R\ge R_0\) . For any \(R_0\le R_1\le R_2\)  and \(X\in T_{R_1}\) , the maximum principle ([1]} Lemma 4.7) implies that \(u_{R_1}^\pm (X)\le u_{R_2}^\pm (X)\le \left\Vert f^\pm \right\Vert _{L^\infty (\mathbb {R}^n)}\) . Therefore, we can define \(u\)  as the monotone limit
\(u(X):=\lim _{R\rightarrow \infty }\left(u_R^+(X)-u_R^-(X)\right) \qquad \forall \, X\in \mathbb {R}^{n+1}_+.\) 
